Abstract

In structural equation modeling, researchers conduct goodness-of-fit tests to evaluate whether the specified model fits the data well. With nonnormal data, the standard goodness-of-fit test statistic T does not follow a chi-square distribution. Comparing T to ${\chi }_{df}^2$ can fail to control Type I error rates and lead to misleading model selection conclusions. To better evaluate model fit, researchers have proposed various robust test statistics, but none of them consistently control Type I error rates under all examined conditions. To improve model fit statistics for nonnormal data, we propose to use an unbiased distribution free weight matrix estimator (${\widehat{\mathbf{\Gamma }}}_{\mathbf{DF}}^{\mathbf{U}}$) in robust test statistics. Specifically, using normal theory based parameter estimates with ${\widehat{\mathbf{\Gamma }}}_{\mathbf{DF}}^{\mathbf{U}},$ we calculate various robust test statistics and robust standard errors. We conducted a simulation study to compare 63 existing robust statistic combinations with the 4 proposed robust statistics with ${\widehat{\mathbf{\Gamma }}}_{\mathbf{DF}}^{\mathbf{U}}.$ The Satorra–Bentler statistic T_SB based on ${\widehat{\mathbf{\Gamma }}}_{\mathbf{DF}}^{\mathbf{U}}$ ($T_{SB}^U$) provided acceptable Type I error rates at $\alpha =.01,$.05, or .1 across all conditions (except a few cases with $\alpha =.01$), regardless of the sample size and the distribution. $T_{SB}^U$ or $T_{\mathit{MVA}2}^U$ typically provided the smallest Anderson-Darling test values, showing the smallest distances between p-values and $\mathit{Uniform}(0,1).$ We use a real data example to compare statistics with ${\widehat{\mathbf{\Gamma }}}_{\mathbf{DF}}^{\mathbf{U}}$ and that with ${\widehat{\mathbf{\Gamma }}}_{\mathbf{ADF}}.$

摘要

在结构方程建模中，研究人员进行拟合优度检验以评估指定模型是否很好地拟合数据。对于非正态数据，标准拟合优度检验统计量T不服从卡方分布。将T与${\chi }_{dF}^{\mathrm{2个}}$可能无法控制 I 类错误率并导致误导性的模型选择结论。为了更好地评估模型拟合度，研究人员提出了各种可靠的测试统计数据，但没有一个能够在所有检查条件下始终如一地控制 I 类错误率。为了改进非正态数据的模型拟合统计，我们建议使用无偏分布自由权重矩阵估计器（${\widehat{\mathbf{\Gamma }}}_{\mathbf{测向仪}}^{\mathbf{ü}}$) 在稳健的测试统计中。具体来说，使用基于正态理论的参数估计${\widehat{\mathbf{\Gamma }}}_{\mathbf{测向仪}}^{\mathbf{ü}},$我们计算各种稳健的测试统计量和稳健的标准误差。我们进行了一项模拟研究，将 63 个现有的稳健统计组合与 4 个提出的稳健统计进行比较${\widehat{\mathbf{\Gamma }}}_{\mathbf{测向仪}}^{\mathbf{ü}}.$Satorra–Bentler 统计量T _SB基于${\widehat{\mathbf{\Gamma }}}_{\mathbf{测向仪}}^{\mathbf{ü}}$(${吨}_{\mathrm{小号}乙}^{ü}$) 提供了可接受的 I 类错误率$\alpha =.01,$.05，或 .1 在所有条件下（除了少数情况$\alpha =.01$)，与样本大小和分布无关。${吨}_{\mathrm{小号}乙}^{ü}$或者${吨}_{\mathit{MVA}\mathrm{2个}}^{ü}$通常提供最小的 Anderson-Darling 检验值，显示p值和$\mathit{制服}(0,\mathrm{1个}).$我们使用一个真实的数据示例来比较统计数据${\widehat{\mathbf{\Gamma }}}_{\mathbf{测向仪}}^{\mathbf{ü}}$那与${\widehat{\mathbf{\Gamma }}}_{\mathbf{自动进纸器}}.$